Engaging students: Deriving the proportions of a 30-60-90 triangle

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Emily Bruce. Her topic, from Geometry: deriving the proportions of a 30-60-90 triangle.

How could you as a teacher create an activity or project that involves your topic?

There is a great activity for deriving the ratio of the sides of a 30-60-90 triangle that uses an equilateral triangle with known side lengths. If you draw the line that bisects one of the angles in the triangle, it is then perpendicular to the side opposite the bisected angle. This creates two triangles with a corresponding 30-degree angle (from the bisected angle), a congruent corresponding side (the line drawn through the triangle), and a corresponding right angle (from the perpendicular line). From this information the two triangles are congruent by the ASA rule. Students might also use the SAS rule by recognizing that the sides of an equilateral triangle are the same lengths, so the two sides adjacent to the bisected 60-degree angle will be congruent. Since the two smaller triangles are congruent, we can show that the smaller sides of the triangle are half the length of the hypotenuse. Using the Pythagorean theorem, the students can find out what the ratio of the sides will be. This is a great activity because it uses students’ prior knowledge about equilateral triangles, angle bisectors, perpendicular lines, and congruent triangles to derive the ratio on their own.

How can this topic be used in your students’ future courses in mathematics and science?

Memorizing the ratio of these sides is not critical in mathematics, because they can always be derived; however having these ratios memorized is very helpful for future use in mathematics and science. When students get into precalculus, they learn about trigonometry. 30-60-90 triangles and their side ratios are specifically helpful when it comes to learning about the unit circle. Students will have to learn the different values of the sine, cosine, and tangent functions of common angles like 30, 60 and 90 that correspond to special right triangles. What they will learn is that for a 30-degree angle, the sine function is equal to the opposite angle divided by the hypotenuse. If the students have memorized the 30-60-90 side ratios, computing these values is simple. Another way in which this can be helpful is in physics. One important topic in physics is projectile motion. In order to find out how far a projectile object will go before it hits the ground, the initial velocity, which is usually at a certain angle upward, must first be split up into its vertical and horizontal components. To do this, they set up the problem as a right triangle, with the initial velocity as the hypotenuse and the angle the object is launched as one of the angles of the triangle. In order to find the vertical and horizontal components of the velocity, it is just a matter of finding the other sides of the triangle. If it so happens that the object was shot at a nice angle like 30 or 60 degrees, students can use their ratio to quickly and easily find the vertical and horizontal components of the velocity.

How can technology be used to effectively engage students with this topic?

A great website for learning and practicing with special right triangles is kahnacademy.org. It provides a video for how to derive the ratios for special right triangles. The way they derive the 30-60-90 ratio is very similar to the activity I described above. This is a great resource for students who may want to go back and look at how the activity was done. The website has many other videos with practice problems. It shows a problem and how to solve it. This gives students a visual example of how to solve some of the questions that might appear on homework. Finally, the website includes word problems and more videos that extend what they students are learning and apply it. The application part of a math topic is extremely important because if students can see the importance of what they’re learning, they will be more inclined to learn it well.